The most common and most likely answer to this crossword clue is the 7 letter word SQUINCH
I've seen this crossword clue in The Guardian
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More clues leading to the result SQUINCH
Buddy explains!

The straight part
We know that "a supporter there" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a supporter there = squinch.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SQUINCH mean?
SQUINCH noun- A structure constructed between two adjacent walls to aid in the transition from a polygonal to a circular structure, as when a dome is constructed on top of a square room.
- To scrunch up (one's face, etc.).